Generalizations of Nekrasov-Okounkov Identity

Abstract

Nekrasov-Okounkov identity gives a product representation of the sum over partitions of a certain function of partition hook length. In this paper we give several generalizations of the Nekrasov-Okounkov identity using the cyclic symmetry of the topological vertex.

Cut-and-Join Operators and Macdonald Polynomials From the 3-Schur Functions

“…Moreover, at level three it can happen that 3-Schur functions are naturally describing the entire multi-parametric set of Kerov functions [15], not only their Macdonal locus (at level two the difference is not seen: generic Kerov in this case is exactly Macdonald). We also did not reveal the relation to triple Macdonald polynomials of [12], which are directly applicable to network model studies [9] -still do not yet allow to formulate the character ∼ character relation [18], what can actually require building a systematic first-principle theory of 3-Schur functions, which is the target of [10] and of the present paper. This, however, can take time and quite some effort.…”

“…Increasing space-time dimension leads to q and t-deformations, substituting Schur by Macdonald polynomials [7]. However, they are not enough for description of generic 6d case, where the single-loop Virasoro is lifted to a double-loop DIM algebra [8], Young diagrams are substituted by plane (3d) partitions, and matrix models are promoted to network models [9], defined on a rich variety of graphs. Needed in this case are the new "3-Schur" functions, depending on additional time variables -they were recently introduced in [10].…”

Cauchy formula and the character ring

“…However, one can wonder what are orthogonality constraints on these p 4 -dependent terms and if they look resolvable. Such analysis in the future can help to find a substitute of (17), which better reflects the structure of plane, rather than ordinary partitions. Coming back to multiplication rules, the differences between expected and actual formulas are marked by boxes.…”

“…One of the many widely-known examples of (2), appears when we integrate exactly over x and y in (1) to obtain a correlator of screenings e φ(x) dx [9]. In this case the a combination of (1) and (2) immediately provides the AGT-induced [13] Nekrasov decomposition [14] of conformal blocks, realized in terms of conformal (Dotsenko-Fateev) matrix models [15] -and their far-going network-model generalizations [17]. Further steps on this way, as well as a related development with matrix −→ tensor model generalizations require essential extension of the theory of Schur characters in various directions.…”

Statistics of close-packed dimers on fractal lattices

“…Although introduced as a simple model for adsorption of diatomic molecules on crystal surfaces, connections with other models in physics and chemistry have been established since then. It has been shown that the close-packed dimer model is equivalent to the two dimensional Ising model [8], and correspondence with many quantum theoretical field theory models has been recognized [9][10][11][12][13]. In graph theory the model is also referred to as perfect matchings and is closely related with other combinatorial objects such as domino tillings [14] and spaning trees [15,16].…”